1ECE 661 - HW 9Abhiram Gnanasambandam

agnanasa@purdue.edu

I. THEORY

A. Fundamental matrix

Fundamental matrix has a 7DoFs. We have to get at least 8 correspondences between the two images toestimate the Fundamental matrix. We have to normalize the pixels tuch thae mean is zero and the avergaedistance from the center is

√2 so the the Hartley and Zisserman algorithm gives better results. Denote the

transformation which does this as T1 and T2.Then using the normalized 8 points,we get an estimate of the F matix, given two sets of corrspondences xi

and x′i/ The theory goes as follows. We have to solve the equation Af = 0, where

A =[x′ixi x

′iyi x

′i y

′ixi y

′iyi xi yi 1

]The matrix gets populated by 8 such rows given by 8 such correspondences.

F =[F11 F12 F13 F21 F22 F23 F31 F32 F33

]TWe can esaily solve for F by using SVD and finding the eigen-vector which corrresponds to the minimum

eigen value. We have to make sure that rank of such a F matrix is always 2. We can ensure it by forcing thethird and least eigen value to go to 0. Then we can denormalize the matrix F, by using Fde = TT2 FT1. Now,we can use the LM optimization to get better estimates.

Now, we have to estimate P and P’ the Projection matrices to construct the world coordinated. As the camerasare not calibrated, we find the canonocal form. P = [I|0] and P ′ = [[e′]xF |e′]

The world co-orrdinates are obtained by finding the solution to AXworld = 0 given the constraint ||Xworld =1||. Here A is given by

A =

xiP

T3 − PT1

yiPT3 − PT2

x′iP′T3 − P ′T1

y′iP′T3 − P ′T2

Now, we generate the error function that will be used to optimize the F. The error function is projecting

the world coordinate using P and P’ to the respective image coordinates and finding the square of euclideandistance with the already existing coordinates. Now, we have to use LM to optimize the estimate of F again.

B. Image Rectification

We first rotate the image 2 by T1 matrix. We then find the angle the eppole makes with x-axis and rotatethe image so that the epipole beecomes parallel to x-axis.

e′ =

f01

The next step os tp send the epipole to infinity. For that we use the following G matrix.

G =

1 0 00 1 0−1/f 0 1

Now, we multiply with another transform T2 to make image back to the center. So, the overall Homography

is H2 = T2GRT1.As we have the homography for Image2, we now have to find the homography for image 1. We use least

squares mnimization to find it.minH1

∑d

(H1xi, H2x′i)

The details of the extended method can be found in Multiple View Geometry by Richard Hartley and AndrewZisserman.

Once, the homographies are found, we can use them to get the rectified images.

C. Using SIFT and canny edge detectors

As we have already done HWs on Canny and SIFT, I am excluding the discussions on them here. We findthe corresponding points between the two rectified images using one of two methods. And once we have tocorresponding methods, we learnt in last two sections how to triangulate and find the world coordinates.

II. RESULTS

Figure 1: View 1

Figure 2: View 2

2

Figure 3: Rectified Images and the SIFT correspondences

Figure 4: Result of canny on rectified image 1

3

Figure 5: Result of canny on rectified image 2

4

Figure 6: The projected 3D points with SIFT on the rectified images

5

Figure 7: The projected 3D points with SIFT on the rectified images , with details for understanding the scenestructure

Figure 8: The projected 3D points with SIFT on the rectified images , before (red lines and black circles) andafter(blue lines and red dots) using LM optimization.

6

Figure 9: The projected 3D points with Canny on the rectified images

7

III. CODE

# −*− co d i ng : u t f −8 −*−" " "C r e a t e d on Sun Nov 18 1 2 : 4 9 : 4 6 2018

@author : a b h i r" " "

i m p o r t cv2 as cvi m p o r t numpy as npfrom s c i p y . o p t i m i z e i m p o r t l e a s t _ s q u a r e s # LM a l g o r i t h m from h e r ei m p o r t mathi m p o r t m a t p l o t l i b . p y p l o t a s p l tfrom m p l _ t o o l k i t s . mplot3d i m p o r t Axes3D

d e f r e t u r n _ c o r r e s p o n d e n c e s ( edges1 , edges2 ) :p1 = [ ]p2 = [ ]f o r i i n r a n g e ( 3 , l e n ( edges1 )−3) :

f o r j i n r a n g e ( 3 , l e n ( edges1 [ 0 ] ) −3 ) :i f ( edges1 [ i , j ] == 2 5 5 ) :

d i s t =1 e10i 1 = if o r j 1 i n r a n g e ( 3 , l e n ( edges2 [0])− 3 ) :

i f ( np . l i n a l g . norm ( im1 [ i −2: i +3 , j −2: j +3] − im2 [ i1 −2: i 1 +3 , j1 −2: j 1 + 3 ] ) < d i s t ) :t = j 1d i s t = np . l i n a l g . norm ( im1 [ i −2: i +3 , j −2: j +3] − im2 [ i1 −2: i 1 +3 , j1 −2: j 1 + 3 ] )

p1 . append ( [ i , j ] )p2 . append ( [ i1 , t ] )

r e t u r n p1 , p2

d e f d r a w _ l i n e s ( image1 , image2 , p o i n t s 1 , p o i n t s 2 , p o i n t s 1 1 , p o i n t s 2 1 , number ) :s1 = np . shape ( image1 )s2 = np . shape ( image2 )

nimage = np . z e r o s ( ( s1 [ 0 ] , s1 [ 1 ] + s2 [ 1 ] , 3 ) ) ;nimage [ : , 0 : s1 [ 1 ] ] = image1nimage [ : , s1 [ 1 ] : ] = image2

# nimage = nimage . a s t y p e ( ’ u i n t 8 ’ )

i f ( number == 0 ) :f o r i i n r a n g e ( 0 , l e n ( p o i n t s 1 ) ) :

cv . l i n e ( nimage , ( p o i n t s 1 [ i ] [ 1 ] , p o i n t s 1 [ i ] [ 0 ] ) , ( s1 [ 1 ] + p o i n t s 2 [ i ] [ 1 ] , p o i n t s 2 [ i ] [ 0 ] ) , ( 0 , 0 , 2 5 5 ) , 1 )

i f ( number == 1 ) :f o r i i n r a n g e ( 0 , l e n ( p o i n t s 1 1 ) ) :

cv . l i n e ( nimage , ( p o i n t s 1 1 [ i ] [ 1 ] , p o i n t s 1 1 [ i ] [ 0 ] ) , ( s1 [ 1 ] + p o i n t s 2 1 [ i ] [ 1 ] , p o i n t s 2 1 [ i ] [ 0 ] ) , ( 0 , 2 5 5 , 0 ) , 1 )

r e t u r n nimage

8

d e f r e t u r n _ c o r r e s _ p o i n t s ( SSd , h a r r i s 1 , h a r r i s 2 , SSDthresh , NCCthresh ) :l 1 = l e n ( SSd )

points1_NCC = [ ]points2_NCC = [ ]

poin ts1_SSD = [ ]poin ts2_SSD = [ ]

p o i n t 2 = [ ]

f o r i i n r a n g e ( 0 , l 1 ) :i f ( np . sum ( SSd [ i ] < SSDthresh ) > 0) :

j 1 = np . argmin ( SSd [ i ] )i f ( j 1 n o t i n p o i n t 2 ) :

i f ( np . abs ( h a r r i s 1 [ i ] [ 0 ] − h a r r i s 2 [ j 1 ] [ 0 ] ) < 3 0 ) :p o i n t 2 . append ( j 1 )poin ts1_SSD . append ( [ h a r r i s 1 [ i ] [ 0 ] , h a r r i s 1 [ i ] [ 1 ] ] )poin ts2_SSD . append ( [ h a r r i s 2 [ j 1 ] [ 0 ] , h a r r i s 2 [ j 1 ] [ 1 ] ] )

r e t u r n points1_SSD , points2_SSD , points1_NCC , points2_NCCd e f p 2 p _ c o r r e s ( des1 , des2 ) :

l 1 = l e n ( des1 )l 2 = l e n ( des2 )

SSD = np . z e r o s ( ( l1 , l 2 ) )

f o r i i n r a n g e ( 0 , l 1 ) :f o r j i n r a n g e ( 0 , l 2 ) :

SSD[ i , j ] = np . l i n a l g . norm ( des1 [ i ] − des2 [ j ] )

r e t u r n SSD

d e f r e c t i f y i m a g e ( im1 , H1 , fn , kkkk ) :Hinv = H1# F i n d i n g t h e s i z e o f t h e o u t p u t imaget l = np . matmul ( Hinv , [ 0 . 0 , 0 . 0 , 1 . 0 ] ) ; #Map f o r Top l e f t c o r n e rt r = np . matmul ( Hinv , [ l e n ( im1 [ 0 ] ) , 0 . 0 , 1 . 0 ] ) ; #Map f o r Top r i g h t c o r n e rb r = np . matmul ( Hinv , [ l e n ( im1 [ 0 ] ) , l e n ( im1 ) , 1 . 0 ] ) ; #Map f o r bot tom r i g h t c o r n e rb l = np . matmul ( Hinv , [ 0 . 0 , l e n ( im1 ) , 1 . 0 ] ) ; #Map f o r bot tom l e f t

t l = t l / t l [ 2 ] ;t r = t r / t r [ 2 ] ;b r = br / b r [ 2 ] ;b l = b l / b l [ 2 ] ;

p r i n t t l , t r , br , b l

9

# Find x l e n g t h o f t h e o u t p u t imageminx = np . min ( [ t l [ 0 ] , t r [ 0 ] , b r [ 0 ] , b l [ 0 ] ] ) ;maxx = np . max ( [ t l [ 0 ] , t r [ 0 ] , b r [ 0 ] , b l [ 0 ] ] ) ;x l e n = np . round ( maxx − minx ) . a s t y p e ( ’ i n t ’ ) ;

# Find y l e n g t h o f t h e o u t p u t imageminy = np . min ( [ t l [ 1 ] , t r [ 1 ] , b r [ 1 ] , b l [ 1 ] ] ) ;maxy = np . max ( [ t l [ 1 ] , t r [ 1 ] , b r [ 1 ] , b l [ 1 ] ] ) ;y l e n = np . round ( maxy − miny ) . a s t y p e ( ’ i n t ’ ) ;

# p r i n t y len , x l e n#### I n i t i a l i s e t h e oupu t image

H _ s c a l i n g = np . eye ( 3 ) ;p r i n t ’ asd ’p r i n t minx , miny , maxx , maxy , x len , y l e n

# minx = −15.5947476171# miny = −138.06632525# maxx = 623.459010284# maxy = 585.306281335# x l e n = 639# y l e n = 723

H _ s c a l i n g [ 0 , 0 ] = f l o a t ( np . shape ( im1 ) [ 1 ] ) / x l e n / 2H _ s c a l i n g [ 1 , 1 ] = f l o a t ( np . shape ( im1 ) [ 0 ] ) / y l e n / 2

Hinv = np . matmul ( H_sca l ing , Hinv )

# p r i n t H _ s c a l i n g / H _ s c a l i n g [ 2 , 2 ]

t l = np . matmul ( Hinv , [ 0 . 0 , 0 . 0 , 1 . 0 ] ) ; #Map f o r Top l e f t c o r n e rt r = np . matmul ( Hinv , [ l e n ( im1 [ 0 ] ) , 0 . 0 , 1 . 0 ] ) ; #Map f o r Top r i g h t c o r n e rb r = np . matmul ( Hinv , [ l e n ( im1 [ 0 ] ) , l e n ( im1 ) , 1 . 0 ] ) ; #Map f o r bot tom r i g h t c o r n e rb l = np . matmul ( Hinv , [ 0 . 0 , l e n ( im1 ) , 1 . 0 ] ) ; #Map f o r bot tom l e f t

t l = t l / t l [ 2 ] ;t r = t r / t r [ 2 ] ;b r = br / b r [ 2 ] ;b l = b l / b l [ 2 ] ;

# p r i n t t l , t r , br , b l# Find x l e n g t h o f t h e o u t p u t imageminx = np . min ( [ t l [ 0 ] , t r [ 0 ] , b r [ 0 ] , b l [ 0 ] ] ) ;maxx = np . max ( [ t l [ 0 ] , t r [ 0 ] , b r [ 0 ] , b l [ 0 ] ] ) ;x l e n = np . round ( maxx − minx ) . a s t y p e ( ’ i n t ’ ) ;

# Find y l e n g t h o f t h e o u t p u t imageminy = np . min ( [ t l [ 1 ] , t r [ 1 ] , b r [ 1 ] , b l [ 1 ] ] ) ;maxy = np . max ( [ t l [ 1 ] , t r [ 1 ] , b r [ 1 ] , b l [ 1 ] ] ) ;y l e n = np . round ( maxy − miny ) . a s t y p e ( ’ i n t ’ ) ;

10

im3 = np . z e r o s ( ( y len , x len , 3 ) ) . a s t y p e ( ’ u i n t 8 ’ ) ;

H1 = np . l i n a l g . i n v ( Hinv )

# p r i n t H1 / H1 [ 2 ] [ 2 ]# Get t h e imagef o r i i n r a n g e ( 0 , l e n ( im3 ) ) :

f o r j i n r a n g e ( 0 , l e n ( im3 [ 0 ] ) ) :newindex = np . matmul ( H1 , [ j + minx , i +miny , 1 . 0 ] )newindex = newindex / newindex [ 2 ] ;

#Use b i l i n e a r a p p r o x i m a t i o n t o g e t t h e o u t p u t p i x e l v a l u ei f ( newindex [ 1 ] > 0 and newindex [ 1 ] < l e n ( im1)−1 and newindex [ 0 ] > 0 and newindex [ 0 ] < l e n ( im1 [ 0 ] ) −1 ) :

f l o o r x = np . f l o o r ( newindex [ 1 ] ) ;f l o o r x i = f l o o r x . a s t y p e ( ’ i n t ’ ) ;f l o o r y = np . f l o o r ( newindex [ 0 ] ) ;f l o o r y i = f l o o r y . a s t y p e ( ’ i n t ’ ) ;

c e i l x = np . c e i l ( newindex [ 1 ] ) ;c e i l x i = c e i l x . a s t y p e ( ’ i n t ’ ) ;c e i l y = np . c e i l ( newindex [ 0 ] ) ;c e i l y i = c e i l y . a s t y p e ( ’ i n t ’ ) ;

p i x e l _ v a l u e = ( c e i l x − newindex [ 1 ] ) * ( c e i l y − newindex [ 0 ] )* im1 [ f l o o r x i ] [ f l o o r y i ] + ( c e i l x − newindex [ 1 ] ) * ( newindex [ 0 ] − f l o o r y )* im1 [ f l o o r x i ] [ c e i l y i ] + ( newindex [ 1 ] − f l o o r x ) * ( newindex [ 0 ] − f l o o r y )* im1 [ c e i l x i ] [ c e i l y i ] + ( newindex [ 1 ] − f l o o r x ) * ( c e i l y − newindex [ 0 ] )* im1 [ c e i l x i ] [ f l o o r y i ] ;

im3 [ i ] [ j ] = p i x e l _ v a l u e . a s t y p e ( ’ u i n t 8 ’ )

# Wr i t e t h e imagecv . i m w r i t e ( fn , im3 )

kkk=np . z e r o s ( ( l e n ( kkkk ) , 3 ) )

f o r i i n r a n g e ( 0 , l e n ( kkkk ) ) :kkk [ i ] = np . matmul ( Hinv , [ kkkk [ i , 0 ] , kkkk [ i , 1 ] , 1 . 0 ] )kkk [ i ] = kkk [ i ] / kkk [ i , 2 ]

kkk [ i , 0 ] = kkk [ i , 0 ] − minxkkk [ i , 1 ] = kkk [ i , 1 ] − miny

r e t u r n im3 , kkk

# d e f r e c t i f y _ i m a g e s ( image , H, p o i n t s ) :

# Find homography f o r r e c t i f i c a t i o nd e f f ind_homo ( e1 , e2 , P1 , P2 , cp1 , cp2 , F1 ) :

x1 = cp1 ;x2 = cp2 ;s s = np . shape ( im1 ) ;

a n g l e = math . a t a n (−( e2 [1]− s s [ 0 ] / 2 ) / ( e2 [0]− s s [ 1 ] / 2 ) )

11

p r i n t a n g l e# p r i n t a n g l e

f = math . cos ( a n g l e ) * ( e2 [0]− s s [ 1 ] / 2 ) −math . s i n ( a n g l e ) * ( e2 [1]− s s [ 0 ] / 2 ) ;# p r i n t f

G = np . z e r o s ( ( 3 , 3 ) )G[ 0 , 0 ] = 1G[ 1 , 1 ] = 1G[ 2 , 0 ] = −1/ fG[ 2 , 2 ] = 1 ;

R = np . z e r o s ( ( 3 , 3 ) )R [ 0 ] [ 0 ] = math . cos ( a n g l e )R[ 0 , 1 ] = −math . s i n ( a n g l e )R[ 1 , 0 ] = math . s i n ( a n g l e )R[ 1 , 1 ] = math . cos ( a n g l e )R[ 2 , 2 ] = 1

T = np . z e r o s ( ( 3 , 3 ) )

T [ 0 , 0 ] = 1 ;T [ 0 , 2 ] = −s s [ 1 ] / 2T [ 1 , 1 ] = 1T [ 1 , 2 ] = −s s [ 0 ] / 2T [ 2 , 2 ] = 1 ;

H2 = np . matmul (G, np . matmul (R , T ) ) ;

im_c = np . z e r o s ( 3 )

im_c [ 0 ] = s s [ 1 ] / 2 ;im_c [ 1 ] = s s [ 0 ] / 2im_c [ 2 ] = 1 ;

c _ r e c = np . matmul ( H2 , im_c )c _ r e c = c _ r e c / c _ r e c [ 2 ]

T2 = np . eye ( 3 )T2 [ 0 , 2 ] = s s [1] /2− c _ r e c [ 0 ]T2 [ 1 , 2 ] = s s [0] /2− c _ r e c [ 1 ]

H2 = np . matmul ( T2 , H2 ) ##########

# F i r s t e p i p o l e# M = np . matmul ( P2 , np . l i n a l g . p inv ( P1 ) )

sec_m = np . ones ( ( 3 , 3 ) )

sec_m [ : , 0 ] = e2 ;sec_m [ : , 1 ] = e2 ;sec_m [ : , 2 ] = e2 ;

12

F1 = F1 / F1 [ 2 , 2 ]

# p r i n t a_x ( e2 ) , e2 , F1M = np . matmul ( a_x ( e2 ) , F1 ) + sec_m# M = exF + e v t

# p r i n t M/M[ 2 , 2 ]

# M = np . matmul ( P2 , np . l i n a l g . p inv ( P1 ) )H0 = np . matmul ( H2 ,M)

# p r i n t H0l = l e n ( cp1 )

x1_dash = np . z e r o s ( ( l , 3 ) ) ;x2_dash = np . z e r o s ( ( l , 3 ) ) ;

f o r i i n r a n g e ( 0 , l ) :x1_dash [ i ] = np . matmul ( H0 , x1 [ i ] )x1_dash [ i ] = x1_dash [ i ] / x1_dash [ i , 2 ]

x2_dash [ i ] = np . matmul ( H2 , x2 [ i ] )x2_dash [ i ] = x2_dash [ i ] / x2_dash [ i , 2 ]

A = x1_dashb = x2_dash [ : , 0 ]

# p r i n t A, b

x = np . matmul ( np . l i n a l g . p inv (A) , b )

Ha = np . eye ( 3 )Ha [ 0 , 0 ] = x [ 0 ]Ha [ 0 , 1 ] = x [ 1 ]Ha [ 0 , 2 ] = x [ 2 ]

H1 = np . matmul ( Ha , H0 )

c _ r e c = np . matmul ( H1 , im_c )c _ r e c = c _ r e c / c _ r e c [ 2 ]

T1 = np . eye ( 3 )T1 [ 0 , 2 ] = s s [1] /2− c _ r e c [ 0 ]T1 [ 1 , 2 ] = s s [0] /2− c _ r e c [ 1 ]

H1 = np . matmul ( T1 , H1 )

F_rec = np . matmul ( np . l i n a l g . p inv ( np . t r a n s p o s e ( H2 ) ) , np . matmul ( F1 , np . l i n a l g . p inv ( H1 ) ) ) ;

x 1_ re c = np . z e r o s ( ( l , 3 ) ) ;x 2_ re c = np . z e r o s ( ( l , 3 ) ) ;

13

f o r i i n r a n g e ( 0 , l ) :x 1_ re c [ i ] = np . matmul ( H1 , x1 [ i ] )x 1_ re c [ i ] = x 1_ re c [ i ] / x 1_ r ec [ i , 2 ]

x 2_ re c [ i ] = np . matmul ( H2 , x2 [ i ] )x 2_ re c [ i ] = x 2_ re c [ i ] / x 2_ r ec [ i , 2 ]

u , s , vh = np . l i n a l g . svd ( F_rec )

e 1 _ r e c = np . t r a n s p o s e ( vh [ 2 ] )e 1 _ r e c = e 1 _ r e c / e 1 _ r e c [ 2 ]e 2 _ r e c = u [ : , 2 ]e 2 _ r e c = e 2 _ r e c / e 2 _ r e c [ 2 ]

r e t u r n x1_rec , x2_rec , H1 , H2 , F_rec , e1_rec , e 2 _ r e c

# LM c o s t f o r F m a t r i xd e f c o s t f ( f ) :

F = np . z e r o s ( ( 3 , 3 ) )F [ 0 , 0 ] = f [ 0 ]F [ 0 , 1 ] = f [ 1 ]F [ 0 , 2 ] = f [ 2 ]F [ 1 , 0 ] = f [ 3 ]F [ 1 , 1 ] = f [ 4 ]F [ 1 , 2 ] = f [ 5 ]F [ 2 , 0 ] = f [ 6 ]F [ 2 , 1 ] = f [ 7 ]F [ 2 , 2 ] = f [ 8 ]

u , s , vh = np . l i n a l g . svd ( F )

# s [ 2 ] = 0## F = np . matmul ( u , np . matmul ( np . d i a g ( s ) , vh ) )# u , s , vh = np . l i n a l g . svd ( F )

e1 = np . t r a n s p o s e ( vh [ 2 ] )

e2 = u [ : , 2 ]

# p r i n t e1 , e2

P = np . z e r o s ( ( 3 , 4 ) )

P [ 0 : 3 , 0 : 3 ] = np . eye ( 3 )

e2x = a_x ( e2 )

P_dash = np . z e r o s ( ( 3 , 4 ) )

P_dash [ 0 : 3 , 0 : 3 ] = np . matmul ( e2x , F )P_dash [ 0 : 3 , 3 ] = e2

# p r i n t e2x# p r i n t ’ \ n \ n \ n ’

14

# p r i n t P , P_dashC = [ ]l = l e n ( cp1 )f o r i i n r a n g e ( 0 , l ) :

A = np . z e r o s ( ( 4 , 4 ) ) ;A[ 0 ] = cp1 [ i ] [ 0 ] * P [ 2 , : ] − P [ 0 , : ]A[ 1 ] = cp1 [ i ] [ 1 ] * P [ 2 , : ] − P [ 1 , : ]A[ 2 ] = cp2 [ i ] [ 0 ] * P_dash [ 2 , : ] − P_dash [ 0 , : ]A[ 3 ] = cp2 [ i ] [ 1 ] * P_dash [ 2 , : ] − P_dash [ 1 , : ]

u , s , vh = np . l i n a l g . svd (A)

xw = np . t r a n s p o s e ( vh [ 3 ] )

xw = xw / np . l i n a l g . norm ( xw )

x1 = np . matmul ( P , xw )x2 = np . matmul ( P_dash , xw )

x1 = x1 / x1 [ 2 ]x2 = x2 / x2 [ 2 ]

# p r i n t x1 , x2# p r i n t np . l i n a l g . norm ( x1 − cp1 ) * * 2 , np . l i n a l g . norm ( x2 − cp2 )**2# p r i n t cp1 , cp2

C . append ( np . l i n a l g . norm ( x1 − cp1 [ i ] ) * * 2 ) ;C . append ( np . l i n a l g . norm ( x2 − cp2 [ i ] ) * * 2 )

# p r i n t np . l i n a l g . norm (C)

r e t u r n np . a s a r r a y (C)

d e f a_x (w ) :wx = w[ 0 ]wy = w[ 1 ]wz = w[ 2 ]Wx = np . z e r o s ( ( 3 , 3 ) )Wx[ 0 ] [ 1 ] = −wzWx[ 0 ] [ 2 ] = wyWx[ 1 ] [ 0 ] = wzWx[ 1 ] [ 2 ] = −wxWx[ 2 , 0 ] = −wyWx[ 2 , 1 ] = wx

r e t u r n Wx# Find t h e Fundamenta l Ma t r i xd e f f i n d F ( np1 , np2 ) :

# S e t he m a t r i c e s f o r s o l v i n g f o r Fl = l e n ( cp1 )A = np . z e r o s ( ( l , 9 ) ) ;

f o r i i n r a n g e ( 0 , l ) :A[ i , 0 ] = np2 [ i ] [ 0 ] * np1 [ i ] [ 0 ]A[ i , 1 ] = np2 [ i ] [ 0 ] * np1 [ i ] [ 1 ]A[ i , 2 ] = np2 [ i ] [ 0 ]A[ i , 3 ] = np2 [ i ] [ 1 ] * np1 [ i ] [ 0 ]

15

A[ i , 4 ] = np2 [ i ] [ 1 ] * np1 [ i ] [ 1 ]A[ i , 5 ] = np2 [ i ] [ 1 ]A[ i , 6 ] = np1 [ i ] [ 0 ]A[ i , 7 ] = np1 [ i ] [ 1 ]A[ i , 8 ] = 1

p r i n t A[ : , 6 ] , np1 [ : , 0 ]u , s , vh = np . l i n a l g . svd (A ) ;

f = np . t r a n s p o s e ( vh [ 8 ] )

F = np . z e r o s ( ( 3 , 3 ) )F [ 0 , 0 ] = f [ 0 ]F [ 0 , 1 ] = f [ 1 ]F [ 0 , 2 ] = f [ 2 ]F [ 1 , 0 ] = f [ 3 ]F [ 1 , 1 ] = f [ 4 ]F [ 1 , 2 ] = f [ 5 ]F [ 2 , 0 ] = f [ 6 ]F [ 2 , 1 ] = f [ 7 ]F [ 2 , 2 ] = f [ 8 ]

u , s , vh = np . l i n a l g . svd ( F ) ;

s [ 2 ] = 0F = np . matmul ( u , np . matmul ( np . d i a g ( s ) , vh ) )

r e t u r n F

# Th i s f u n c t i o n n o r m a l i z e s t h e p o i n t sd e f n o r m a l i z e _ p o i n t s ( p o i n t s ) :

m1 = np . mean ( p o i n t s [ : , 0 ] ) ;m2 = np . mean ( p o i n t s [ : , 1 ] ) ;

p r i n t np . shape ( p o i n t s ) , np . shape (m1)

d i s t 1 = ( p o i n t s [ : , 0 ] −m1 ) * ( p o i n t s [ : , 0 ] −m1)d i s t 2 = ( p o i n t s [ : , 1 ] −m2 ) * ( p o i n t s [ : , 1 ] −m2)

d i s t = np . z e r o s ( l e n ( p o i n t s ) )

f o r i i n r a n g e ( 0 , l e n ( p o i n t s ) ) :d i s t [ i ] = np . s q r t ( d i s t 1 [ i ] + d i s t 2 [ i ] )

md = np . sum ( d i s t ) / l e n ( p o i n t s )

s c a l e = np . s q r t ( 2 ) / md ;

n e w _ p o i n t s = np . ones ( np . shape ( p o i n t s ) )

T = np . z e r o s ( ( 3 , 3 ) )

T [ 0 ] [ 0 ] = s c a l eT [ 0 ] [ 2 ] = −s c a l e * m1T [ 1 ] [ 1 ] = s c a l e ;

16

T [ 1 ] [ 2 ] = −s c a l e * m2 ;T [ 2 ] [ 2 ] = 1 ;

f o r i i n r a n g e ( 0 , l e n ( p o i n t s ) ) :n e w _ p o i n t s [ i ] [ 0 ] = ( p o i n t s [ i ] [ 0 ] − m1)* s c a l en e w _ p o i n t s [ i ] [ 1 ] = ( p o i n t s [ i ] [ 1 ] − m2)* s c a l e

# new_po in t s1 = np . ones ( np . shape ( p o i n t s ) )## f o r i i n r a n g e ( 0 , 8 ) :# new_po in t s1 [ i ] = np . matmul ( T , p o i n t s [ i ] ) ;

r e t u r n new_poin t s , T

im1 = cv . imread ( ’ 1 1 . png ’ )im2 = cv . imread ( ’ 1 2 . png ’ )

cp1 = np . z e r o s ( ( 8 , 3 ) )cp2 = np . z e r o s ( ( 8 , 3 ) )

# cp1 [ 0 ] = [ 1 2 8 6 , 2 1 5 8 , 1 ]# cp1 [ 1 ] = [ 1 7 3 1 , 2 6 2 1 , 1 ]# cp1 [ 2 ] = [ 2 2 6 1 , 2 5 0 3 , 1 ]# cp1 [ 3 ] = [ 2 3 1 7 , 1 3 2 0 , 1 ]# cp1 [ 4 ] = [ 1 7 8 5 , 1 3 2 6 , 1 ]# cp1 [ 5 ] = [ 1 3 0 3 , 1 2 3 7 , 1 ]# cp1 [ 6 ] = [ 1 4 9 2 , 1 6 2 6 , 1 ]# cp1 [ 7 ] = [ 1 4 9 7 , 1 3 8 9 , 1 ]#### cp2 [ 0 ] = [ 1 3 6 9 , 2 1 2 0 , 1 ]# cp2 [ 1 ] = [ 1 7 8 6 , 2 5 8 4 , 1 ]# cp2 [ 2 ] = [ 2 3 2 2 , 2 4 6 9 , 1 ]# cp2 [ 3 ] = [ 2 3 6 8 , 1 2 8 0 , 1 ]# cp2 [ 4 ] = [ 1 8 2 5 , 1 2 9 1 , 1 ]# cp2 [ 5 ] = [ 1 3 7 4 , 1 2 0 3 , 1 ]# cp2 [ 6 ] = [ 1 5 5 4 , 1 5 9 2 , 1 ]# cp2 [ 7 ] = [ 1 5 5 8 , 1 3 5 8 , 1 ]

cp1 [ 0 ] = [ 9 4 , 2 3 1 , 1 ]cp1 [ 1 ] = [ 3 8 4 , 4 1 5 , 1 ]cp1 [ 2 ] = [ 4 2 , 1 2 3 , 1 ]cp1 [ 3 ] = [ 3 8 4 , 3 1 1 , 1 ]cp1 [ 4 ] = [ 2 0 5 , 1 3 , 1 ]cp1 [ 5 ] = [ 5 3 5 , 1 2 1 , 1 ]cp1 [ 6 ] = [ 5 1 0 , 2 4 4 , 1 ]cp1 [ 7 ] = [ 1 8 6 , 1 4 7 , 1 ]

cp2 [ 0 ] = [ 1 0 0 , 2 1 2 , 1 ]cp2 [ 1 ] = [ 3 3 1 , 4 1 9 , 1 ]cp2 [ 2 ] = [ 6 0 , 1 0 7 , 1 ]

17

cp2 [ 3 ] = [ 3 1 8 , 3 1 6 , 1 ]cp2 [ 4 ] = [ 2 2 8 , 2 2 , 1 ]cp2 [ 5 ] = [ 5 1 0 , 1 5 4 , 1 ]cp2 [ 6 ] = [ 4 9 0 , 2 6 8 , 1 ]cp2 [ 7 ] = [ 1 7 6 , 1 4 1 , 1 ]

#Img1 = [ ]#Img2 = [ ]#Img1 . append ( [ 1 8 5 9 . 0 , 4 4 7 . 0 , 1 . 0 ] )#Img1 . append ( [ 3 9 0 5 . 0 , 1 0 1 1 . 0 , 1 . 0 ] )#Img1 . append ( [ 4 8 1 . 0 , 8 9 7 . 0 , 1 . 0 ] )#Img1 . append ( [ 3 0 0 1 . 0 , 2 0 3 3 . 0 , 1 . 0 ] )#Img1 . append ( [ 7 3 7 . 0 , 2 0 9 3 . 0 , 1 . 0 ] )#Img1 . append ( [ 2 7 0 1 . 0 , 3 3 4 1 . 0 , 1 . 0 ] )#Img1 . append ( [ 2 0 7 3 . 0 , 1 9 3 7 . 0 , 1 . 0 ] )#Img1 . append ( [ 2 7 7 7 . 0 , 2 7 8 5 . 0 , 1 . 0 ] )#####Img2 . append ( [ 1 9 2 1 . 0 , 4 5 9 . 0 , 1 . 0 ] )#Img2 . append ( [ 3 9 7 7 . 0 , 9 8 5 . 0 , 1 . 0 ] )#Img2 . append ( [ 5 5 7 . 0 , 8 5 3 . 0 , 1 . 0 ] )#Img2 . append ( [ 3 0 9 3 . 0 , 1 9 2 9 . 0 , 1 . 0 ] )#Img2 . append ( [ 7 9 3 . 0 , 2 0 7 7 . 0 , 1 . 0 ] )#Img2 . append ( [ 2 7 6 9 . 0 , 3 3 2 9 . 0 , 1 . 0 ] )#Img2 . append ( [ 2 1 5 3 . 0 , 1 8 6 9 . 0 , 1 . 0 ] )#Img2 . append ( [ 2 8 5 7 . 0 , 2 7 2 5 . 0 , 1 . 0 ] )## cp1 = np . a s a r r a y ( Img1 )# cp2 = np . a s a r r a y ( Img2 )

kkkk1 = cp1 [ 0 : 7 ] ;kkkk2 = cp2 [ 0 : 7 ] ;

f o r i i n r a n g e ( 0 , 8 ) :cv . c i r c l e ( im1 , ( i n t ( cp1 [ i , 0 ] ) , i n t ( cp1 [ i , 1 ] ) ) , 1 , ( 0 , 0 , 2 5 5 ) , −1)cv . c i r c l e ( im2 , ( i n t ( cp2 [ i , 0 ] ) , i n t ( cp2 [ i , 1 ] ) ) , 1 , ( 0 , 0 , 2 5 5 ) , −1)

# n o r m a l i z e d p o i n t snp1 , T1 = n o r m a l i z e _ p o i n t s ( cp1 )np2 , T2 = n o r m a l i z e _ p o i n t s ( cp2 )

Fde = f i n d F ( np1 , np2 )

F = np . matmul ( np . t r a n s p o s e ( T2 ) , np . matmul ( Fde , T1 ) ) ;

F = F / F [ 2 , 2 ]u , s , vh = np . l i n a l g . svd ( F )

e1 = np . t r a n s p o s e ( vh [ 2 ] )e2 = u [ : , 2 ]

P = np . z e r o s ( ( 3 , 4 ) )

18

P [ 0 : 3 , 0 : 3 ] = np . eye ( 3 )

e2x = a_x ( e2 )

P_dash = np . z e r o s ( ( 3 , 4 ) )

P_dash [ 0 : 3 , 0 : 3 ] = np . matmul ( e2x , F )P_dash [ 0 : 3 , 3 ] = e2F = F / F [ 2 , 2 ]f = np . z e r o s ( 9 )f [ 0 ] = F [ 0 ] [ 0 ]f [ 1 ] = F [ 0 ] [ 1 ]f [ 2 ] = F [ 0 ] [ 2 ]f [ 3 ] = F [ 1 ] [ 0 ]f [ 4 ] = F [ 1 ] [ 1 ]f [ 5 ] = F [ 1 ] [ 2 ]f [ 6 ] = F [ 2 ] [ 0 ]f [ 7 ] = F [ 2 ] [ 1 ]f [ 8 ] = F [ 2 ] [ 2 ]

# r e s = l e a s t _ s q u a r e s ( c o s t f , f , method = ’ lm ’ )# f o r i i n r a n g e ( 0 , 8 )f1 = f

F1 = np . z e r o s ( ( 3 , 3 ) )F1 [ 0 , 0 ] = f1 [ 0 ]F1 [ 0 , 1 ] = f1 [ 1 ]F1 [ 0 , 2 ] = f1 [ 2 ]F1 [ 1 , 0 ] = f1 [ 3 ]F1 [ 1 , 1 ] = f1 [ 4 ]F1 [ 1 , 2 ] = f1 [ 5 ]F1 [ 2 , 0 ] = f1 [ 6 ]F1 [ 2 , 1 ] = f1 [ 7 ]F1 [ 2 , 2 ] = f1 [ 8 ]

F1 = F1 / F1 [ 2 , 2 ]

#u , s , vh = np . l i n a l g . svd ( F1 )## s [ 2 ] = 0##F1 = np . matmul ( u , np . matmul ( np . d i a g ( s ) , vh ) )#u , s , vh = np . l i n a l g . svd ( F1 )e1 = np . t r a n s p o s e ( vh [ 2 ] )e2 = u [ : , 2 ]

P = np . z e r o s ( ( 3 , 4 ) )

P [ 0 : 3 , 0 : 3 ] = np . eye ( 3 )

e1 = e1 / e1 [ 2 ]e2 = e2 / e2 [ 2 ]

19

e2x = a_x ( e2 )

P_dash = np . z e r o s ( ( 3 , 4 ) )P_dash [ 0 : 3 , 0 : 3 ] = np . matmul ( e2x , F1 )P_dash [ 0 : 3 , 3 ] = e2

x1_rec , x2_rec , H1 , H2 , F_rec , e1_rec , e 2 _ r e c = find_homo ( e1 , e2 , P , P_dash , cp1 , cp2 , F1 )# x1_rec , x2_rec , H1 , H2 , F_rec , e1_rec , e 2 _ r e c = find_homo ( e1 , e2 , P , P_dash , cp1 , cp2 )

i m a g e 1 c l r , kkk1 = r e c t i f y i m a g e ( im1 , H1 , ’ r e c 1 . png ’ , kkkk1 )i m a g e 2 c l r , kkk2 = r e c t i f y i m a g e ( im2 , H2 , ’ r e c 2 . png ’ , kkkk2 )

kkk1 = kkk1 . a s t y p e ( ’ i n t ’ )kkk2 = kkk2 . a s t y p e ( ’ i n t ’ )#SIFT# s i f t = cv . x f e a t u r e s 2 d . S I F T _ c r e a t e ( )# i m a g e 1 c l r = cv . r e s i z e ( i m a g e 1 c l r , ( 0 , 0 ) , fx = 0 . 2 , fy = 0 . 2 )# image1gray = cv . c v t C o l o r ( i m a g e 1 c l r , cv .COLOR_BGR2GRAY)##kp1 , des1 = s i f t . de tec tAndCompute ( image1gray , None )## h a r r i s 1 = [ ]## f o r i i n kp1 :# h a r r i s 1 . append ( [ i . p t [ 1 ] , i . p t [ 0 ] ] )#### i m a g e 2 c l r = cv . imread ( ’m4 . jpg ’ )## i m a g e 2 c l r = cv . r e s i z e ( i m a g e 2 c l r , ( 0 , 0 ) , fx = 0 . 2 , fy = 0 . 2 )# image2gray = cv . c v t C o l o r ( i m a g e 2 c l r , cv .COLOR_BGR2GRAY)#kp2 , des2 = s i f t . de tec tAndCompute ( image2gray , None )## h a r r i s 2 = [ ]## f o r i i n kp2 :# h a r r i s 2 . append ( [ i . p t [ 1 ] , i . p t [ 0 ] ] )##SSD = p 2 p _ c o r r e s ( des1 , des2 )### f o r i i n r a n g e ( 1 , 1 0 0 0 ) :## i f ( np . sum (SSD < i ) >= 1 0 0 0 ) :## l l = i## b r e a k ;## points1_SSD , points2_SSD , points1_NCC , points2_NCC = r e t u r n _ c o r r e s _ p o i n t s ( SSD , h a r r i s 1 , h a r r i s 2 , 2 0 0 , 0 . 9 9 )## poin ts1_SSD = np . a s a r r a y ( poin ts1_SSD ) . a s t y p e ( ’ i n t ’ )# poin ts2_SSD = np . a s a r r a y ( poin ts2_SSD ) . a s t y p e ( ’ i n t ’ )# points1_NCC = np . a s a r r a y ( poin ts1_SSD ) . a s t y p e ( ’ i n t ’ )# points2_NCC = np . a s a r r a y ( poin ts2_SSD ) . a s t y p e ( ’ i n t ’ )#### nimage1 = d r a w _ l i n e s ( i m a g e 1 c l r , i m a g e 2 c l r , points1_SSD , points2_SSD , points1_NCC , points2_NCC , 0 )## s t r i = ’ s i f t . jpg ’

20

# cv . i m w r i t e ( s t r i , nimage1 )

#Cannyimage1gray = cv . c v t C o l o r ( i m a g e 1 c l r , cv .COLOR_BGR2GRAY)edges1 = cv . Canny ( image1gray , 2 5 5 * 1 . 5 , 2 5 5 )

image2gray = cv . c v t C o l o r ( i m a g e 2 c l r , cv .COLOR_BGR2GRAY)edges2 = cv . Canny ( image2gray , 2 5 5 * 1 . 5 , 2 5 5 )

points1_SSD , poin ts2_SSD = r e t u r n _ c o r r e s p o n d e n c e s ( edges1 , edges2 ) ;

poin ts1_SSD = np . a s a r r a y ( poin ts1_SSD ) . a s t y p e ( ’ i n t ’ )poin ts2_SSD = np . a s a r r a y ( poin ts2_SSD ) . a s t y p e ( ’ i n t ’ )points1_NCC = np . a s a r r a y ( poin ts1_SSD ) . a s t y p e ( ’ i n t ’ )points2_NCC = np . a s a r r a y ( poin ts2_SSD ) . a s t y p e ( ’ i n t ’ )

nimage1 = d r a w _ l i n e s ( i m a g e 1 c l r , i m a g e 2 c l r , points1_SSD , points2_SSD , points1_NCC , points2_NCC , 0 )

s t r i = ’ s i f t . jpg ’cv . i m w r i t e ( s t r i , nimage1 )

#############Img1 = [ ]Img2 = [ ]f o r i i n r a n g e ( l e n ( poin ts1_SSD ) ) :

Img1 . append ( [ poin ts1_SSD [ i ] [ 0 ] , po in ts1_SSD [ i ] [ 1 ] , 1 . 0 ] )Img2 . append ( [ poin ts2_SSD [ i ] [ 0 ] , po in ts2_SSD [ i ] [ 1 ] , 1 . 0 ] )

f o r j i n r a n g e ( l e n ( kkk1 ) ) :Img1 . append ( [ kkk1 [ j ] [ 1 ] , kkk1 [ j ] [ 0 ] , 1 . 0 ] )Img2 . append ( [ kkk2 [ j ] [ 1 ] , kkk2 [ j ] [ 0 ] , 1 . 0 ] )

cp1 = np . a s a r r a y ( Img1 )cp2 = np . a s a r r a y ( Img2 )

# f o r i i n r a n g e ( 0 , 8 ) :# cv . c i r c l e ( im1 , ( i n t ( cp1 [ i , 0 ] ) , i n t ( cp1 [ i , 1 ] ) ) , 1 , ( 0 , 0 , 2 5 5 ) , −1)# cv . c i r c l e ( im2 , ( i n t ( cp2 [ i , 0 ] ) , i n t ( cp2 [ i , 1 ] ) ) , 1 , ( 0 , 0 , 2 5 5 ) , −1)

# n o r m a l i z e d p o i n t snp1 , T1 = n o r m a l i z e _ p o i n t s ( cp1 )np2 , T2 = n o r m a l i z e _ p o i n t s ( cp2 )

Fde = f i n d F ( np1 , np2 )

F = np . matmul ( np . t r a n s p o s e ( T2 ) , np . matmul ( Fde , T1 ) ) ;

F = F / F [ 2 , 2 ]u , s , vh = np . l i n a l g . svd ( F )

e1 = np . t r a n s p o s e ( vh [ 2 ] )e2 = u [ : , 2 ]

21

P = np . z e r o s ( ( 3 , 4 ) )

P [ 0 : 3 , 0 : 3 ] = np . eye ( 3 )

e2x = a_x ( e2 )

P_dash = np . z e r o s ( ( 3 , 4 ) )

P_dash [ 0 : 3 , 0 : 3 ] = np . matmul ( e2x , F )P_dash [ 0 : 3 , 3 ] = e2F = F / F [ 2 , 2 ]f = np . z e r o s ( 9 )f [ 0 ] = F [ 0 ] [ 0 ]f [ 1 ] = F [ 0 ] [ 1 ]f [ 2 ] = F [ 0 ] [ 2 ]f [ 3 ] = F [ 1 ] [ 0 ]f [ 4 ] = F [ 1 ] [ 1 ]f [ 5 ] = F [ 1 ] [ 2 ]f [ 6 ] = F [ 2 ] [ 0 ]f [ 7 ] = F [ 2 ] [ 1 ]f [ 8 ] = F [ 2 ] [ 2 ]

# r e s = l e a s t _ s q u a r e s ( c o s t f , f , method = ’ lm ’ )# f o r i i n r a n g e ( 0 , 8 )f1 = f

F1 = np . z e r o s ( ( 3 , 3 ) )F1 [ 0 , 0 ] = f1 [ 0 ]F1 [ 0 , 1 ] = f1 [ 1 ]F1 [ 0 , 2 ] = f1 [ 2 ]F1 [ 1 , 0 ] = f1 [ 3 ]F1 [ 1 , 1 ] = f1 [ 4 ]F1 [ 1 , 2 ] = f1 [ 5 ]F1 [ 2 , 0 ] = f1 [ 6 ]F1 [ 2 , 1 ] = f1 [ 7 ]F1 [ 2 , 2 ] = f1 [ 8 ]

F1 = F1 / F1 [ 2 , 2 ]

#u , s , vh = np . l i n a l g . svd ( F1 )## s [ 2 ] = 0##F1 = np . matmul ( u , np . matmul ( np . d i a g ( s ) , vh ) )#u , s , vh = np . l i n a l g . svd ( F1 )e1 = np . t r a n s p o s e ( vh [ 2 ] )e2 = u [ : , 2 ]

P = np . z e r o s ( ( 3 , 4 ) )

P [ 0 : 3 , 0 : 3 ] = np . eye ( 3 )

22

e1 = e1 / e1 [ 2 ]e2 = e2 / e2 [ 2 ]e2x = a_x ( e2 )

P_dash = np . z e r o s ( ( 3 , 4 ) )P_dash [ 0 : 3 , 0 : 3 ] = np . matmul ( e2x , F1 )P_dash [ 0 : 3 , 3 ] = e2

xw = np . z e r o s ( ( l e n ( cp1 ) , 3 ) )f o r i i n r a n g e ( 0 , l e n ( cp1 ) ) :

A = np . z e r o s ( ( 4 , 4 ) ) ;A[ 0 ] = cp1 [ i ] [ 0 ] * P [ 2 , : ] − P [ 0 , : ]A[ 1 ] = cp1 [ i ] [ 1 ] * P [ 2 , : ] − P [ 1 , : ]A[ 2 ] = cp2 [ i ] [ 0 ] * P_dash [ 2 , : ] − P_dash [ 0 , : ]A[ 3 ] = cp2 [ i ] [ 1 ] * P_dash [ 2 , : ] − P_dash [ 1 , : ]

u , s , vh = np . l i n a l g . svd (A)

xw1 = np . t r a n s p o s e ( vh [ 3 ] )xw1 = xw1 / xw1 [ 3 ]

xw [ i ] = xw1 [ 0 : 3 ]

f i g = p l t . f i g u r e ( )ax = f i g . a d d _ s u b p l o t ( 1 1 1 , p r o j e c t i o n = ’3d ’ )ax . s c a t t e r ( xw [ : , 0 ] , xw [ : , 1 ] , xw [ : , 2 ] )# ax . p l o t ( xw [ 5 0 : 5 2 , 0 ] , xw [ 5 0 : 5 2 , 1 ] , xw [ 5 0 : 5 2 , 2 ] )# ax . p l o t ( xw [ 5 2 : 5 4 , 0 ] , xw [ 5 2 : 5 4 , 1 ] , xw [ 5 2 : 5 4 , 2 ] )# ax . p l o t ( xw [ 5 4 : 5 6 , 0 ] , xw [ 5 4 : 5 6 , 1 ] , xw [ 5 4 : 5 6 , 2 ] )# ax . p l o t ( [ xw [ 5 0 , 0 ] , xw [ 5 2 , 0 ] ] , [ xw [ 5 0 , 1 ] , xw [ 5 2 , 1 ] ] , [ xw [ 5 0 , 2 ] , xw [ 5 2 , 2 ] ] )# ax . p l o t ( [ xw [ 5 2 , 0 ] , xw [ 5 4 , 0 ] ] , [ xw [ 5 2 , 1 ] , xw [ 5 4 , 1 ] ] , [ xw [ 5 2 , 2 ] , xw [ 5 4 , 2 ] ] )# ax . p l o t ( [ xw [ 5 1 , 0 ] , xw [ 5 3 , 0 ] ] , [ xw [ 5 1 , 1 ] , xw [ 5 3 , 1 ] ] , [ xw [ 5 1 , 2 ] , xw [ 5 3 , 2 ] ] )# ax . p l o t ( [ xw [ 5 3 , 0 ] , xw [ 5 5 , 0 ] ] , [ xw [ 5 3 , 1 ] , xw [ 5 5 , 1 ] ] , [ xw [ 5 3 , 2 ] , xw [ 5 5 , 2 ] ] )

23

TheoryFundamental matrix Image RectificationUsing SIFT and canny edge detectors

Resultscode